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September 11, 2006

Wirth and Stasheff on Homotopy Transition Cocycles

Posted by Urs Schreiber

Way back in 1965, James Wirth wrote a PhD thesis on the description of fibrations

(1)p:EB p : E \to B

in terms of transition data

(2)p 1(U α) t α t¯ α U α×F p p U α = U α. \array{ p^{-1}(U_\alpha) & \array{\stackrel{t_\alpha}{\rightarrow} \\ \stackrel{\bar t_\alpha}{\leftarrow}} & U_\alpha \times F \\ p \downarrow \;\; && \;\;\downarrow p \\ U_\alpha &=& U_\alpha } \,.

Here U αU_\alpha are elements of a good covering of BB by open sets, FF is the typical fiber, t αt_\alpha is a chosen trivialization of the fibration over U αU_\alpha, and t¯ α\bar t_\alpha its inverse, up to homotopy.

A new arXiv entry now recalls the main idea of this old work in modern language:

James Wirth & Jim Stasheff
Homotopy Transition Cocycles

The situation looks a lot like that familiar from the local trivialization of principal fiber bundles. The crucial generalization, though, is in the very last clause, which asserts that t αt_\alpha and t¯ α\bar t_\alpha are only weak inverses of each other.

As a result of that, the familiar cocycle equation

(1)g αβg βγ=g αγ g_{\alpha\beta }g_{\beta \gamma} = g_{\alpha \gamma}

for the transition functions

(2)g αβ=t¯ αt β g_{\alpha\beta} = \bar t_\alpha t_\beta

will in general only hold up to homotopy

(3)g αβg βγf αβγg αγ. g_{\alpha\beta }g_{\beta \gamma} \stackrel{f_{\alpha\beta\gamma}}{\simeq} g_{\alpha \gamma} \,.

If these f f_{\cdots} satisfy an equation postulating a sort of associativity, we get a higher cocycle equation as known from gerbes. But, still more generally, this f f_\cdots itself might be associative only up to homotopy, and so on. This yields cocycles of the form as they corespond to 2-gerbes, 3-gerbes, etc.

One may neatly wrap up all this information in terms of what I would call a pseudofunctor

(4)UC, \mathbf{U} \to C \,,

where UU is what I know as the Čech groupoid of the good covering αU α\sqcup_\alpha U_\alpha and where CC is some nn-category. In the topological world this is called a functor up to strong homotopy.

(For some pictures of how such functors look like I can point for instance to this and this. The general idea of realizing general cocycles as functors from simplices to certain codomains was also formulated by John E. Roberts.)

Wirth worked out to what extent fibrations are equivalent to their collection of transition data (“descent data”).

James F. Wirth
Fiber spaces and the higher homotopy cocycle relations
PhD thesis, University of Notre Dame, 1965

The difficult part is to reconstruct the fibration EE from the transitions between its local trivializations.

For a fiber bundle, we just take the space αU α×F\sqcup_\alpha U_\alpha \times F and identify points which are related under g αβg_{\alpha\beta}. Obviously, for general fibrations this construction is more involved, since the g αβg_{\alpha\beta} are far from inducing an equivalence relation.

The crucial tool for making progress is the mapping cylinder theorem which was stated and proven in

James F. Wirth
The mapping cylinder axiom for WCHP fibrations
Pac. J. Math. 54(2):263-279, 1974 .

I thank Jim Stasheff for making me aware of this work in the comment section of this entry. I might have a comment and a question, but I will post these to the comment section of this entry here.

Posted at September 11, 2006 7:15 PM UTC

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Re: Wirth and Stasheff on Homotopy Transition Cocycles

The following is the announced comment, or rather some hybrid between a comment and a question:

John Baez and Toby Bartels have investigated the idea of studying such issues in terms of nn-bundles .

An nn-bundle is a (topological or smooth, depending on your application) nn-category EE together with a suitable functor

(1)p:EB, p : E \to B \,,

where BB is a discrete nn-category, the base space.

In terms of such nn-bundles there are more of less obvious categorifications of many of the structures familar from ordinary gauge theory, like principal bundles, vector bundles, connections on bundles, etc.

Just as in the theory of homotopy transition cocycles, it is relatively easy to pass from a given nn-bundle structure EBE \to B to the corresponding cocycle data of transitions between local trivializations.

What is not at all obvious is the inverse of this construction. I am not aware that anyone has tried to address this question in the context of nn-bundles. But possibly some of Wirth’s work may be applied here.

Personally, I was already puzzled for quite a while by what should be the simplest nontrivial example.

Given a Deligne 3-cocycle on a base space BB, i.e. the cocycle describing a U(1)U(1)-gerbe with connection and curving on BB, I wanted to know which total 2-space EBE \to B with connection

(2)tra:P 2(B)Trans(B) \mathrm{tra} : P_2(B) \to \mathrm{Trans}(B)

has local trivialization data that gives rise to the specified Deligne 3-cocycle.

My approach to this question was probably hopelessly unsophisticated as compared to Wirth’s tools. But the solution that I finally came up with at least has a form that suits rather nicely the purpose of such a description in the context where I wanted to apply it to, namely the “physics” of strings in Kalb-Ramond backgrounds.

The solution that I came up with is described here.

Roughly, the idea is this:

To the U(1)U(1)-gerbe we may associate a PU(H)PU(H) principal bundle, and to that we may associate a vector bundle whose fibers are algebras A xA_x of compact operators on some Hilbert space.

Then from the connection and curving data we find a transport 2-functor, which sends points xx to algebras A xA_x, paths xγyx \stackrel{\gamma}{\to} y to A xA_x-A yA_y bimdodules and cobordisms between “parallel” paths to bimodule homomorphisms.

Now, this yields a total space which is not a category, but just a set (with extra structure).

But the 2-category (= bicategory, I always say nn-category for the weakest possible case) of bimodules naturally sits inside the 2-category Mod Vect\mathrm{Mod}_\mathrm{Vect} of module categories for the monoidal category Vect\mathrm{Vect}.

Under this map

(3)BimMod Vect \mathrm{Bim} \to \mathrm{Mod}_\mathrm{Vect}

an algebra A xA_x is sent to the category of left A xA_x-modules (which is itself a module category for Vect\mathrm{Vect}, by tensoring from the right), a bimodule is sent to a functor between categories of left modules, and so on.

This way we may think of each of our fibers A xA_x as actually denoting a category, namely A xMod{}_{A_x}\mathrm{Mod}.

This point of view is nice because it manifestly realizes a line bundle gerbe as a 2-vector bundle in the sense that objects of Mod Vect\mathrm{Mod}_\mathrm{Vect} can be addressed as 2-vector spaces #.

On the other hand, it is not quite so nice, because the total “2-space”

(4)E:= x A xMod E := \sqcup_x {}_{A_x}\mathrm{Mod}

is not a topological (much less a smooth) category. At least not without further work and further assumptions.

So this modest observation of mine is the comment I have. The question of course is if maybe Wirth’s result could be helpful for understanding such issues of 2-bundles.

Posted by: urs on September 11, 2006 8:40 PM | Permalink | Reply to this
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